A060210 -id:A060210 - cp's OEIS frontend

A063983 Least k such that k*2^n +/- 1 are twin primes.

4, 2, 1, 9, 12, 6, 3, 9, 57, 30, 15, 99, 165, 90, 45, 24, 12, 6, 3, 69, 132, 66, 33, 486, 243, 324, 162, 81, 90, 45, 345, 681, 585, 375, 267, 426, 213, 429, 288, 144, 72, 36, 18, 9, 147, 810, 405, 354, 177, 1854, 927, 1125, 1197, 666, 333, 519, 1032, 516, 258, 129, 72

Offset: 0

Robert G. Wilson v, Sep 06 2001

nonn

Excluding the first three terms, all remaining terms have digital root 3, 6, or 9. - J. W. Helkenberg, Jul 24 2013

			a(3) = 9 because 9*2^3 = 72 and 71 and 73 are twin primes.
a(6) = 3 because 3*2^6 = 192 and {191, 193} are twin primes.
a(71) = 630 because 630*2^71 = 1487545442103938242314240 and {1487545442103938242314239, 1487545442103938242314241} are twin primes.

Richard Crandall and Carl Pomerance, 'Prime Numbers: A Computational Perspective,' Springer-Verlag, NY, 2001, page 12.

Pierre CAMI, Table of n, a(n) for n = 0..2300

Cf. A040040, A045753, A002822, A124065, A124518-A124522.
Cf. A071256, A060210, A060256. For records see A125848, A125019.
Cf. A076806 (requires odd k).

Table[Do[s=(2^j)*k; If[PrimeQ[s-1]&&PrimeQ[s+1],Print[{j,k}]], {k,1,2*j^2}],{j,0,100}]; (* outprint of a[j]=k *)
Do[ k = 1; While[ ! PrimeQ[ k*2^n + 1 ] || ! PrimeQ[ k*2^n - 1 ], k++ ]; Print[ k ], {n, 0, 50} ]
f[n_] := Block[{k = 1},While[Nand @@ PrimeQ[{-1, 1} + 2^n*k], k++ ];k];Table[f[n], {n, 0, 60}] (* Ray Chandler, Jan 09 2009 *)

More terms from Labos Elemer, May 24 2002

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A158034 Integers n for which f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) is an integer.

Original entry on oeis.org

3, 11, 23, 83, 131, 179, 191, 239, 243, 251, 359, 419, 431, 443, 491, 659, 683, 719, 743, 891, 911, 1019, 1031, 1103, 1223, 1439, 1451, 1499, 1511, 1539, 1559, 1583, 1811, 1931, 2003, 2039, 2063, 2211, 2339, 2351, 2399, 2459, 2511, 2543, 2699, 2819, 2903

Offset: 1

Reikku Kulon, Mar 11 2009

Superset of A002515; 2n + 1 is prime. A recursive search for members of this sequence results in the infinite series of very large primes A145918. Most members of this sequence are also prime, but five members less than 10000 are composite:
.. . 243 = 3^5
.. . 891 = 3^4 * 11
. . 1539 = 3^4 * 19
. . 2211 = 3 * 11 * 67
. . 2511 = 3^4 * 31
The polygonal number with f sides of length 2n + 1 is (2^n - 1)(2^(n - 1)).
Contribution from Reikku Kulon, May 19 2009: (Start)
The average difference between successive composite terms gradually increases, remaining near their order of magnitude. Roughly 3% of all primes less than 20 billion belong to this sequence or the 2n + 1 sequence. The interval between composite terms 12228632879 and 13169544651 contains 1113606 primes, accounting for 2.75% of the primes in the interval and 1.42% of the primes between 24457265759 and 26339089303.
Prime factors are most often congruent to 3 (mod 4), but some factors are congruent to 1 (mod 4), especially when a term has an even number of not necessarily distinct factors. The most common factor is 3, and often a large power of 3 is a divisor. 5, 7, 13, and 17 are never factors.
The ones digit of composite terms is most often 1, and becomes progressively more likely to be 1. It is never 5. It cannot be 7, because 2n + 1 would then be divisible by 5. The lack of solutions with n divisible by 5 appears crucial to the consistent primality of 2n + 1.
The tens digit is odd if the ones digit is 1 or 9; it is even if the ones digit is 3. This is a consequence of congruence to 3 (mod 4).
The most common least significant two digits of composite terms are 51.
The least significant digits of prime terms do not follow an obvious distribution.
This is the simplest and possibly most productive member of a family of similar sequences defined by f = (s + 8n^2 - 2) / (2n * (2n + 1)), where s is pronic. For these sequences, 2n + 1 is dominated by primes.
=====================================
Large sequences of consecutive primes
=====================================
.   Initial term    Primes   Predecessor     Successor          Gap
.   ---------------------------------------------------------------
.     1529648303    157285    1529648231    1639846391    110198160
.     3832649339    473045    3832647111    4193496803    360849692
.     5897103683    411434    5897102751    6223464171    326361420
.     6543227423    445293    6543226251    6899473631    356247380
.     8126586971    913506    8126586711    8871331491    744744780
.     9533381219    689395    9533380131   10103115231    569735100
.    11576086883    708712   11576086731   12171829419    595742688
.    12228633251   1113606   12228632879   13169544651    940911772
.    21315457451   2328623   21315457251   23375077119   2059619868
(End)

			ngon(f, k) = k * (f * (k - 1) / 2 - k + 2)
. . . 3 = (4^3 - 2^3 + 8 * 9 - 2) / (6 * 7)
. . . . = (2 * 28 + 70) / 42
. . 126 = (2 * 28 + 70)
.. . 28 = (2^3 - 1) * 2^2
. . . . = 126 - 70 - 28
. . . . = 7 * (18 - 10 - 4)
. . . . = 7 * (3 * 6 - 3 * 3 - 5)
. . . . = 7 * (3 * 3 - 7 + 2)
.. 8287 = (4^11 - 2^11 + 8 * 121 - 2) / (22 * 23)
. . . . = (2 * 2096128 + 966) / 506
4193222 = (2 * 2096128 + 966)
2096128 = (2^11 - 1) * 2^10
. . . . = 4193222 - 2096128 - 966
. . . . = 23 * (182314 - 91136 - 42)
. . . . = 23 * (8287 * 22 - 8287 * 11 - 21)
. . . . = 23 * (8287 * 11 - 23 + 2)
Coincidentally, 8287 = 129 * 64 + 31 = 257 * 32 + 63 is prime, and may be the largest value of f that is.
1031 = 257 * 4 + 3 and 2063 = 1031 * 2 + 1 are both members of this sequence, 4127 = 2063 * 2 + 1 is prime, and 8287 = (4127 + 16) * 2 + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 112.

Cf. A002515 (Lucasian primes)
Cf. A145918 (exponential Sophie Germain primes)
Cf. A139601 (polygonal numbers)
Cf. A046318, A139876 (related to composite members 243, 891, 1539, and 2511)
Cf. A060210, A002034, A109833, A136801 (their factors)
Cf. A039506 (3, 8287)
Cf. A006516 (2^n - 1)(2^(n - 1))
Cf. A000051 (Fermat numbers), A019434 (Fermat primes)
Cf. A142291 (prime sequence 257, 1031, 2063, 4127)
Cf. A235540 (nonprimes), A002943.

a158034 n = a158034_list !! (n-1)
a158034_list = [x | x <- [1..],
                    (4^x - 2^x + 8*x^2 - 2) `mod` (2*x*(2*x + 1)) == 0]
-- Reinhard Zumkeller, Jan 12 2014

A071407 Least k such that kprime(n) + 1 and kprime(n) - 1 are twin primes.

Original entry on oeis.org

2, 2, 6, 6, 18, 24, 6, 12, 6, 12, 42, 54, 30, 24, 6, 120, 18, 258, 24, 18, 84, 132, 54, 48, 114, 42, 6, 6, 48, 24, 144, 30, 6, 12, 12, 78, 24, 36, 30, 54, 132, 18, 90, 36, 66, 18, 42, 30, 120, 30, 36, 42, 18, 18, 54, 84, 60, 12, 210, 12, 6, 60, 150, 102, 6, 210, 30, 24, 6

Offset: 1

Labos Elemer, May 24 2002

Note that 6 divides a(n) for n > 2. - T. D. Noe, Jan 07 2013

			n=4: prime(4)=7, a(4)=6 because 6*prime(4)=42 and {41,43} are primes.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A071256, A071404-A071406, A060256, A060210, A090530, A294731.
Cf. A071558 (k at every integer).
Cf. A220141, A220142 (record values).

a071407 n = head [k | k <- [2,4..], let x = k * a000040 n,
                      a010051' (x - 1) == 1, a010051' (x + 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

Table[fl=1; Do[s=(Prime[j])*k; If[PrimeQ[s-1]&&PrimeQ[s+1]&&Equal[fl, 1], Print[{j, k}]; fl=0], {k, 1, 2*j^2}], {j, 0, 100}]

From Amiram Eldar, Aug 25 2025: (Start)
a(n) = A090530(n) / prime(n).
a(n) = 6 * A294731(n) for n >= 3. (End)

A118701 Largest prime power that divides the average of twin-prime pairs.

Original entry on oeis.org

4, 3, 4, 9, 5, 7, 5, 9, 17, 27, 23, 25, 9, 64, 11, 19, 16, 27, 47, 13, 29, 7, 27, 11, 29, 19, 25, 103, 107, 11, 81, 137, 23, 13, 49, 17, 43, 25, 59, 13, 128, 41, 71, 43, 31, 11, 17, 121, 19, 31, 67, 81, 139, 283, 41, 149, 16, 313, 23, 25, 37, 169, 347, 29, 64

Offset: 1

Greg Huber, Jul 13 2006

nonn

			a(10) = 27 since the 10th twin-prime pair is (107, 109) and 108 = 4*27.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A014574, A034699, A060210, A246655.

Max[Power @@@ FactorInteger[#]] & /@ (Select[Prime[Range[350]], PrimeQ[# + 2] &] + 1) (* Amiram Eldar, May 25 2024 *)

maxpow(n) = {my(f = factor(n)); if(n == 1, 1, vecmax(vector(#f~, i, f[i,1]^f[i,2])));}
lista(pmax) = {my(prev = 2); forprime(p = 3, pmax, if(p == prev + 2, print1(maxpow(p - 1), ", ")); prev = p);} \\ Amiram Eldar, May 25 2024

a(n) = A034699(A014574(n)). - Amiram Eldar, May 25 2024

More terms from Amiram Eldar, May 25 2024

A178668 Maximal prime divisor of the average of the twin prime pairs, different from 2 and 3. In case of maximal prime divisor is 2 or 3, then a(n)=1.

Original entry on oeis.org

1, 1, 1, 1, 5, 7, 5, 1, 17, 1, 23, 5, 5, 1, 11, 19, 5, 5, 47, 13, 29, 7, 1, 11, 29, 19, 5, 103, 107, 11, 5, 137, 23, 13, 7, 17, 43, 7, 59, 13, 1, 41, 71, 43, 31, 11, 17, 11, 19, 31, 67, 5, 139, 283, 41, 149, 13, 313, 23, 13, 37, 13, 347, 29, 11, 71, 17, 373, 7, 11, 13, 397, 17, 1, 443, 7, 113, 13, 31, 467, 11, 5, 13, 11, 271, 181, 11, 37, 7, 281, 113, 577, 17, 7, 59, 593, 199, 17, 157, 13

Offset: 1

Vladimir Shevelev, Dec 25 2010

nonn

78 from the first 100 terms are first or second members of twin pairs and only 12 are not. In a natural supposition that for large prime terms the latter should be in the majority, there are reasons to assume that the number N for which it occurs for the first time is very large.

The average of a twin-prime pair is the same as 1 + the lower twin prime, whose largest prime factor is tabulated in A060210.

Cf. A178456, A178527, A001359

s = Plus @@@ Select[ Partition[ Prime@ Range@ 350, 2, 1], #[[1]] + 2 == #[[2]] &]; f[n_] := Max[First /@ FactorInteger@ n] /. {2 -> 1, 3 -> 1}, f /@ s

cp's OEIS Frontend

A063983 Least k such that k*2^n +/- 1 are twin primes.

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A158034 Integers n for which f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) is an integer.

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A071407 Least k such that k*prime(n) + 1 and k*prime(n) - 1 are twin primes.

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A118701 Largest prime power that divides the average of twin-prime pairs.

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A178668 Maximal prime divisor of the average of the twin prime pairs, different from 2 and 3. In case of maximal prime divisor is 2 or 3, then a(n)=1.

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A071407 Least k such that kprime(n) + 1 and kprime(n) - 1 are twin primes.